The freezing point of water is 32°F. If it rains at a temperature lower than that, it will snow. However, let's say we dig a deep hole not filled with water that goes like 10,000 feet below sea level. The pressure will be greater there, but if the temperature is high enough to cause the rain to boil, what happens to the rain? Does it just evaporate before hitting the ground?
Pressure and temperature are not all the same when going up vertically. See average temperature and pressure profiles. The evaporation of any liquid has a pressure-dependent equilibrium point, see the phase diagram of water and Clausius-Clapeyron for a mathematical description of the equilibrium point.
However, I assume that you mean with 212°F (100°C) the temperature at 1 bar level. Then water would tend to evaporate as it goes through the surrounding hot atmosphere. How quickly this happens depends on the temperature of the droplets and their size, i.e. the conditions of their formation.
If the free-fall speed is higher than the evaporation speed, then droplets would still hit the ground.
Of course this is a hypothetical scenario, as if the ground is at 100°C, then the atmospheric layers where droplets could form will be some 20-30°C colder (so minimum at 160°F). Which means they will not form in the first place.
Which means it would not rain if you assume the ground temperature is 212°F.
It will depend on how the temperature and humidity change as the raindrops approach the bottom of the column. If the air is hot and dry, the drops will evaporate. If it's hot and humid enough, they may survive to hit the bottom. Remember that the boiling point of water rises as pressure increases!
Your statement "If it rains at a temperature lower than [32 F], it will snow" is a bit inaccurate. It can still rain at colder temperatures if there's a warmer layer of air aloft. (This can originate as normal rain, or as snow that melts as it falls through warmer air.) Conversely, it can snow when the ground-level temperature is above freezing, as long as the temperature aloft is below freezing. Finally, we have sleet (rain which freezes as it falls through colder air) and graupel (supercooled water that freezes onto snowflakes, forming little fluffy ice balls).
You can also get freezing rain, which is supercooled liquid water falling as rain -- it's liquid, but below the freezing point at ambient pressure. This rain will freeze when it hits a solid object, creating a layer of ice.
By analogy to this last process, there could be superheated water that still falls as liquid, but bursts into steam as soon as it hits a surface. We wouldn't see that under normal conditions, because it would be falling through air that isn't saturated, so it would undergo evaporative cooling. In your deep, hot, and eventually humid hole, though, it might well be possible -- I'm not sure what would produce such superheating, but I can't rule it out.